I am looking for a short proof of this fact. This is clearly true by drawing these trees, but I am having trouble putting it into writing. Somehow I need to select 3 of the 5 vertices and show that there must be a $3$-cycle or an independent set of size $3$. (Is this a Ramsey number or something like that?)

Is there an obvious/short argument here, or perhaps a lemma I could use to knock it out quickly?

I might be missing something here but this seems too simple. Take a bipartite coloring of your forest and take the largest color cover.
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EuYuNov 24 '12 at 19:47

How is it possible to have cycles in a forest ?
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AmrNov 24 '12 at 19:56

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@EuYu All you're missing is that I'm bad at graph theory. :) That is a great short proof.
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Alexander Gruber♦Nov 24 '12 at 20:23

@EuYu Can your method with color coverings be generalized to graphs of finite girth? For example, could it be used to show that the only graphs with girth $4$ without an independent set of size $3$ is the $4$-cycle?
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Alexander Gruber♦Nov 24 '12 at 20:41

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The argument is dependent more on the fact that forests are bipartite than on their girth. I'm not sure it will generalize nicely for girths. The most general result I could give is that if a graph of order $n$ is $k$-chromatic, then there is an independent set of at least $\frac{n}{k}$. This is a well-known trivial bound on the independence number.
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EuYuNov 24 '12 at 20:53

1 Answer
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In general let $F$ be a forest of order $n$ with $k$ connected components. It follows that every connected component is a tree, thus every component is bipartite (because trees do not have odd cycles). Thus the ith connected component contains an independent set of size greater than ceiling($n_i/2$) (Where $n_i$ is the order of the ith connected component) . Since the union of all these independent sets is again independent (because they come from different connected components), thus we have:

$$\sum_{i=1}^k ceiling(n_i/2)\leq \alpha(F)$$

A weaker version of this inequality would be:
$$n/2=\sum_{i=1}^k (n_i/2)\leq \alpha(F)$$

Since the order of $F$ in your question is 5, we know that $5/2\leq \alpha(F)$ hence $3\leq \alpha(F)$